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In this work the correspondence between the semiclassical limit of the DOZZquantum Liouville theory on the torus and the Nekrasov-Shatashvili limit of theN=2* (Omega-deformed) U(2) super-Yang-Mills theory is used to propose newformulae for the accessory parameter of the Lame equation. This quantity is inparticular crucial for solving the problem of uniformization of theone-punctured torus. The computation of the accessory parameters for torus andsphere is an open longstanding problem which can however be solved if onesucceeds to derive an expression for the so-called classical Liouville action.The method of calculation of the latter has been proposed some time ago byZamolodchikov brothers. Studying the semiclassical limit of the four-pointfunction of the quantum Liouville theory on the sphere they have derived theclassical action for the Riemann sphere with four punctures. In the presentwork Zamolodchikovs idea is exploited in the case of the Liouville field theoryon the torus. It is found that the Lame accessory parameter is determined bythe classical Liouville action on the one-punctured torus or more concretely bythe torus classical block evaluated on the saddle point intermediate classicalweight. Secondly, as an implication of the aforementioned correspondence it isobtained that the torus accessory parameter is related to the sum of allrescaled column lengths of the so-called "critical" Young diagrams extremizingthe instanton "free energy" for the N=2* gauge theory. Finally, it is pointedout that thanks to the known relation the sum over the "critical" columnlengths can be expressed in terms of a contour integral in which the integrandis built out of certain special functions.

Classical torus conformal block, N=2* twisted superpotential and the accessory parameter of Lame equation